A165773 -id:A165773 - cp's OEIS frontend

A055487 Least m such that phi(m) = n!.

1, 3, 7, 35, 143, 779, 5183, 40723, 364087, 3632617, 39916801, 479045521, 6227180929, 87178882081, 1307676655073, 20922799053799, 355687465815361, 6402373865831809, 121645101106397521, 2432902011297772771, 51090942186005065121, 1124000727844660550281, 25852016739206547966721, 620448401734814833377121, 15511210043338862873694721, 403291461126645799820077057, 10888869450418352160768000001, 304888344611714964835479763201

Offset: 1

Labos Elemer, Jun 28 2000

nonn

Erdős believed (see Guy reference) that phi(x) = n! is solvable.
Factorial primes of the form p = A002981(m)! + 1 = k! + 1 give the smallest solutions for some m (like m = 1,2,3,11) as follows: phi(p) = p-1 = A002981(m)!.
According to Tattersall, in 1950 H. Gupta showed that phi(x) = n! is always solvable. - Joseph L. Pe, Oct 01 2002
A123476(n) is a solution to the equation phi(x)=n!. - T. D. Noe, Sep 27 2006
From M. F. Hasler, Oct 04 2009: (Start)
Conjecture: Unless n!+1 is prime (i.e., n in A002981), a(n)=pq where p is the least prime > sqrt(n!) such that (p-1) | n! and q=n!/(p-1)+1 is prime.
Probably "least prime > sqrt(n!)" can also be replaced by "largest prime <= ceiling(sqrt(n!))". The case "= ceiling(...)" occurs for n=5, sqrt(120) = 10.95..., p=11, q=13.
a(n) is the first element in row n of the table A165773, which lists all solutions to phi(x)=n!. Thus a(n) = A165773((Sum_{kA055506(k)) + 1). The last element of each row (i.e., the largest solution to phi(x)=n!) is given in A165774. (End)

R. K. Guy, (1981): Unsolved problems In Number Theory, Springer - page 53.
Tattersall, J., "Elementary Number Theory in Nine Chapters", Cambridge University Press, 2001, p. 162.

Max A. Alekseyev, Computing the Inverses, their Power Sums, and Extrema for Euler's Totient and Other Multiplicative Functions. Journal of Integer Sequences, Vol. 19 (2016), Article 16.5.2.
P. Erdős and J. Lambek, Problem 4221, Amer. Math. Monthly, 55 (1948), 103.

Cf. A055486, A055488, A055489, A055506, A000010, A000142.

Cf. A123476, A165773, A165774.

Array[Block[{k = 1}, While[EulerPhi[k] != #, k++]; k] &[#!] &, 10] (* Michael De Vlieger, Jul 12 2018 *)

a(n) = Min{m : phi(m) = n!} = Min{m : A000010(m) = A000142(n)}.

More terms from Don Reble, Nov 05 2001

a(21)-a(28) from Max Alekseyev, Jul 09 2014

A055506 Number of solutions to the equation phi(x) = n!.

Original entry on oeis.org

2, 3, 4, 10, 17, 49, 93, 359, 1138, 3802, 12124, 52844, 182752, 696647, 2852886, 16423633, 75301815, 367900714, 1531612895, 8389371542, 40423852287, 213232272280, 1295095864798, 7991762413764, 42259876674716, 252869570952706, 1378634826630301, 8749244047999717

Offset: 1

Labos Elemer, Jun 29 2000

nonn

Note that if phi(x) = n!, then x must be a product of primes p such that p - 1 divides n!. - David Wasserman, Apr 30 2002

Gives the row lengths of the table A165773 (see example). All solutions to phi(x)=n! are in the interval [n!,(n+1)!] with the smallest/largest solutions given in A055487/A165774 respectively. - M. F. Hasler, Oct 04 2009

			n = 5, phi(x) = 5! = 120 holds for the following 17 numbers: { 143, 155, 175, 183, 225, 231, 244, 248, 286, 308, 310, 350, 366, 372, 396, 450, 462 }.
From _M. F. Hasler_, Oct 04 2009: (Start)
The table A165773 looks as follows:
  1,2, (a(1)=2 numbers for which phi(n) = 1! = 1)
  3,4,6, (a(2)=3 numbers for which phi(n) = 2! = 2)
  7,9,14,18, (a(3)=4 numbers for which phi(n) = 3! = 6)
  35,39,45,52,56,70,72,78,84,90, (a(4)=10 numbers for which phi(n) = 4! = 24)
  ... (End)

Andrew Lelechenko, Table of n, a(n) for n = 1..36
Max A. Alekseyev, Computing the Inverses, their Power Sums, and Extrema for Euler's Totient and Other Multiplicative Functions. Journal of Integer Sequences, Vol. 19 (2016), Article 16.5.2.
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000010, A000142, A000203, A014197, A054873, A055486, A067847, A165773, A165774.

a(n) = invphiNum(n!); \\ Amiram Eldar, Nov 15 2024 using Max Alekseyev's invphi.gp

use ntheory ":all"; print "$ ",scalar(inverse_totient(factorial($))),"\n" for 1..20; # Dana Jacobsen, Mar 04 2019

a(n) = A014197(n!) = Cardinality({x; A000010(x) = A000142(n)}).

More terms from Jud McCranie, Jan 02 2001
More terms from David Wasserman, Apr 30 2002 (with the assistance of Vladeta Jovovic and Sascha Kurz)
a(21)-a(28) from Max Alekseyev, Jan 26 2012, Jul 09 2014

A165774 Largest solution to phi(x) = n!, where phi() is Euler totient function (A000010).

Original entry on oeis.org

2, 6, 18, 90, 462, 3150, 22050, 210210, 1891890, 19969950, 219669450, 2847714870, 37020293310, 520843112790, 7959363061650, 135309172048050, 2300255924816850, 41996101027370490, 797925919520039310, 16504589035937252250, 347097774991217099850, 7751850308137181896650, 179602728970220622816750, 4493489228616853106091450, 112337230715421327652286250, 2958213742172761628176871250, 79871771038664563960775523750, 2279417465795734863803670716250

Offset: 1

M. F. Hasler, Oct 04 2009

nonn

All solutions to phi(x) = n! belong to the interval [n!,(n+1)!] and are listed in the n-th row of A165773 (when written as table with row lengths A055506). Thus this sequence gives the last element in these rows, i.e., a(n) = A165773(Sum_{k=1..n} A055506(k)).
All terms in this sequence are even, since if x is an odd solution to phi(x) = n!, then 2x is a larger solution because phi(2x) = phi(2)*phi(x) = phi(x).
Most terms (and any term divisible by 4) are divisible by 3, since if x = 2^k*y is a solution with k>1 and gcd(y,2*3) = 1, then x*3/2 = 2^(k-1)*3*y is a larger solution because phi(2^(k-1)*3) = 2^(k-2)*(3-1) = 2^(k-1) = phi(2^k).
For the same reason, most terms are divisible by 5, since if x=2^k*y is a solution with k>2 and gcd(y,2*5) = 1, then x*5/4 is a larger solution.
Also, any term of the form x = 2^k*3^m*y with k,m>1 must be divisible by 7 (else x*7/6 would be a larger solution), and so on.
Experimentally, a(n) = c(n)*(n+1)! with a coefficient c(n) ~ 2^(-n/10) (e.g., c(1) = c(2) = 1, c(10) ~ 0.5).

			a(1) = 2 is the largest among the A055506(1) = 2 solutions {1,2} to phi(n) = 1! = 1.
a(4) = 90 is the largest among the A055506(4) = 10 solutions {35, 39, 45, 52, 56, 70, 72, 78, 84, 90} to phi(n) = 4! = 24.
See A165773 for more examples.

Max A. Alekseyev, Computing the Inverses, their Power Sums, and Extrema for Euler's Totient and Other Multiplicative Functions. Journal of Integer Sequences, Vol. 19 (2016), Article 16.5.2.
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000010, A055487, A055506, A055489, A014197, A165773.

a(n) = invphiMax(n!); \\ Amiram Eldar, Nov 11 2024, using Max Alekseyev's invphi.gp

Edited and terms a(12)-a(28) added by Max Alekseyev, Jan 26 2012, Jul 09 2014

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A055487 Least m such that phi(m) = n!.

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A055506 Number of solutions to the equation phi(x) = n!.

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A165774 Largest solution to phi(x) = n!, where phi() is Euler totient function (A000010).

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